A Novel Technique of Securing Images using Chaos and EZW

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II. CHAOS AND EZW
A. CHAOS BASED SCRAMBLING ALGORITHM:
“Chaos” means “state of disorder”, in mathematics for a system to be called as chaotic system; it should satisfy the following conditions [6],
1. It should be highly sensitive to initial condition.
2. It should satisfy the mixing property.
3. Periodic orbits should be dense.
A chaotic map is a function which exhibits chaotic behavior; there are different types of chaotic maps (like 1-D maps, 2-D maps, discrete-time parameter, continuous-time parameter etc.). In this paper we are considering chaotic logistic map (1-D) which is simple to implement.
Chaotic logistic map is given by equation (1),
Xj+1 =u*Xj*(1-Xj), u [1, 4] & X (0, 1) (1)
Where, u is the bifurcation parameter, if u [3.57, 4] then the equation behaves chaotically, here we are taking ‘u’ value as 4.X0 is the initial value which can be taken randomly from (0, 1).
The Scrambling algorithm used in this paper reorders the rows and columns of the 2-D data. For scrambling rows and columns we are using two different keys.
Scrambling algorithm:
The chaotic logistic map produces chaotic sequence within the range (0, 1). A position vector ‘P’ is generated, which signifies the new positions of the row or column of a given 2-D data.
P = [ Pi, Pi+1… PN ] (2)
Where,
Pi are the position elements.
The position vector is generated based on mixing property of chaotic dynamical systems. The mixing property is given as [7]:
“For any two open intervals I and J (which can be arbitrarily small, but must have a nonzero length) one can find initial values in I which when iterated will eventually lead to points in J”.
Hence mixing property states that any initial condition at any interval can traverse intervals in the ent...
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...tions, P. Gupta, D. Huang, P. Premaratne & X. Zhang, Ed. Berlin: Springer, 2012, pp.259-263.
[6] http://en.wikipedia.org/wiki/Chaos_theory(dated: 28/01/2014)
[7] Heinz-Otto Pentigen, Hartmut Jurgens, Dietmar Saupe, “Chaos and Fractals-New frontiers of science, Second Edition”.
[8] Muhammad Asif Khan, Mohammad Asim, Varun Jeoti, Rana Shahid Manzoor, “Chaos based constellation scrambling in OFDM systems: Security & Interleaving issues”, Information Technology, 2008. ITsim 2008, International Symposium.
[9] Jerome M Shapiro, “Embedded Image coding using zero trees of wavelet co-efficient”, IEEE Transactions on signal processing, VOL. 41, No.12, December 1993.
[10] Amir Said, William A. Pearlman, “A New, Fast and Efficient Image codec based on set partitioning in Hierarchical Trees”, IEEE Transactions on Circuits and systems for video technology, VOL. 6, No.3, June 1996